A ballistic trajectory represents the motion of an object where the only substantive force acting on the object is the earth's gravity. An object is typically placed in such a trajectory through the use of one or more booster rockets that throw the object away from the earth and its atmosphere and into a desired trajectory intended to cause the object to reach a given destination. The booster rocket(s) provide the initial force acting on the object, but once they burn out the object is only acted upon by gravity and it assumes its ballistic trajectory. Such a trajectory is elliptical in shape, occurring fully within a geometric plane that passes through the earth's center. If the ellipse is of sufficient size, the object will continue to orbit the earth in that trajectory indefinitely. If, as is the case with offensive ballistic missiles, the orbit is not sufficiently large for the object to freely orbit the earth, it will collide with the earth at some point in the orbit, but it may do so at a great distance from where it is launched, and it will collide with the earth at enormous speed.
It is believed that Johannes Kepler made the first accurate descriptions of such trajectories in the seventeenth century, the physics of which were then explained later in that century by Isaac Newton. From their work, it is possible to identify six invariants of ballistic trajectories that may be used to uniquely characterize a given trajectory, with these invariants being constant as long as the object orbits the earth. Using such a characterization, it is possible to analyze these trajectories in meaningful ways, such as determining whether different observers far removed from one another are observing the same object.
When tracking objects exhibiting ballistic motion, it is frequently of interest to compare one object's motion to that of another object, or to compare one or more such trajectories to a set of criteria used to identify objects meeting the criteria. The analysis of trajectories is, in general, complicated by their dynamic nature. The comparison of one ballistic trajectory to another, or the comparison of multiple objects to a set of criteria, is problematic since the motion of the objects is non-linear. These non-linearities include variations in speed and course. While it is possible to use the invariants to calculate these parameters for any given position or time, which calculations are used to good effect for much important analysis of such trajectories, the process is cumbersome. For example, for many analyses of potential interest, many such calculations must be performed iteratively to achieve the desired result, particularly when the analysis consists of analyzing many ballistic objects simultaneously. The use of the invariants simplifies the comparison analysis essentially by transforming the analysis in a way that changes it from a dynamic to a static analysis.
In addition, because of the dynamic nature of the trajectories, it is common practice to individually estimate each object's position at some given time, and then compare their estimated positions to determine whether they satisfy the criteria. However, individually estimating and comparing trajectories is inefficient. For example, under the common practice, the work of comparing a given object to all other objects is proportional to the number of objects being observed. If a comparison is required of all objects to all other objects, the comparison effort grows exponentially with the number of objects.
Alternative systems and methods for rapid and accurate determination and analysis of ballistic trajectories are desired.